The density function of the solution of a two-point boundary value problem containing small stochastic processes

Xia, Ning Mao

doi:10.1090/qam/934679

Author: Ning Mao Xia
Journal: Quart. Appl. Math. 46 (1988), 29-47
MSC: Primary 34B15; Secondary 34E05, 34F05, 35R99, 60H10
DOI: https://doi.org/10.1090/qam/934679
MathSciNet review: 934679
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Abstract: This paper concerns a two-point boundary value problem for an $m$thorder system of ordinary differential equations containing a vector stochastic process \[ \xi \left ( {t, \omega } \right ) = {\xi _0}\left ( t \right ) + \varepsilon {\xi _1}\left ( {t, \omega } \right ) + {\varepsilon ^2}{\xi _2}\left ( {t, \omega } \right ) + \cdot \cdot \cdot .\] When $\varepsilon$ is small, the existence and the asymptotic properties of the solution can be obtained by means of the shooting method, and its density function can be determined by solving a sequence of first-order deterministic partial differential equations.

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